erasure of information under conservation laws

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Context Energy cost Angular Mtm cost Impact Summary

Erasure of information under conservation laws

Joan VaccaroCentre for Quantum DynamicsGriffith University Brisbane, Australia

Steve BarnettSUPAUniversity of StrathclydeGlasgow, UK

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Landauer erasure

Landauer, IBM J. Res. Develop. 5, 183 (1961)

00

1

forward process:

0 0

1 0

time reversed:

?

Erasure is irreversible

Minimum cost

00/1

BEFORE erasure AFTER erasure

env2 smicrostate # total N

)2ln( )ln( env kTNkT

)2ln(kTQ

# microstates

environment

)ln( envNkTQ

heat

)2ln( envNkTQ

Context

Hide the past of the memory in a reservoir (who’s past is unknown)

?

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Exorcism of Maxwell’s demon

1871 Maxwell’s demon extracts work of Q from thermal reservoir by collecting only hot gas particles. (Violates 2nd Law: reduces entropy of whole gas)

Q

Thermodynamic Entropy

1982 Bennet showed full cycle requires erasure of demon’s memory which costs at least Q :

Bennett, Int. J. Theor. Phys. 21, 905 (1982)

Cost of erasure is commonly expressed as entropic cost:

This is regarded as the fundamental cost of erasing 1 bit. BUT this result is implicitly associated with an energy cost:

)2ln(kS

STQ

Qwork

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Different Paradigm all states are degenerate in energy maximisation of entropy subject to

conservation of angular momentum cost of erasure is angular momentum

Conventional Paradigm maximisation of entropy subject to

conservation of energy cost of erasure is work

S

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Example to set the stage… single-electron atoms with ground state spin angular momentum memory: spin-1/2 atoms in equal mixture reservoir: spin-1 atoms all in mj = 1 state (spin polarised)

independent optical trapping potentials (dipole traps) atoms exchange spin angular momentum via collisions when traps brought together erasure of memory by loss of spin polarisation of reservoir – the cost of erasure is spin angular momentum

1/2 1/2

1 1 1 1 1 1

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1/2 1/2

1 1 1 1 1 1

Example to set the stage… single-electron atoms with ground state spin angular momentum memory: spin-1/2 atoms in equal mixture reservoir: spin-1 atoms all in mj = 1 state (spin polarised)

independent optical trapping potentials (dipole traps) atoms exchange spin angular momentum via collisions when traps brought together erasure of memory by loss of spin polarisation of reservoir – the cost of erasure is spin angular momentum

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Impact

This talk

Energy CostConventional paradigm:▀ conservation of energy▀ simple 2-state atomic model

New paradigm:▀ conservation of angular momentum ▀ energy degenerate states of different spin

Angular Momentum Cost

▀ New mechanism▀ statements of the 2nd Law

zJ

2

0

1dEE

zJ

,11,0

Shannon

cost work

entropy

E

thermal reservoir spin reservoir

Proc. R. Soc. A 467 1770 (2011)

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System:

0 1 0/1

Memory bit: 2 degenerate atomic states

Thermal reservoir: multi-level atomic gas at temperature T

Z

eP

kTE

E

/

E

Energy cost

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T

Z

eP

kTE

E

/

0 1

Thermalise memory bit while increasing energy gap

0/1

2

11 P

2

10 P

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T

Z

eP

kTE

E

/

raise energy of state(e.g. Stark or Zeeman shift) 0

1dE

0/1

1

dEPdW 1

kTE

kTE

e

eP

/

/

11

kTEeP

/01

1

Work to raise state from E to E+dE


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T

Z

eP

kTE

E

/

0/1

dEPdW 1

01 P


/

/0 0

1log 2

log 21

E kT

E kTE E

eW P dE dE kT

e

Total work

1

0

10 P

raise energy of state(e.g. Stark or Zeeman shift)

1


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T

Z

eP

kTE

E

/

0/1

dEPdW 1

01 P


/

/0 0

1log 2

log 21

E kT

E kTE E

eW P dE dE kT

e

Total work

1

0

10 P

raise energy of state(e.g. Stark or Zeeman shift)

1


Thermalisation of memory bit:

Bring the system to thermal equilibrium at each step in energy:i.e. maximise the entropy of the system subject to conservation of energy.

This is erasure in the paradigm of thermal reservoirs

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• an irreversible process

• based on random interactions to bring the system to maximum entropy subject to a conservation law

• the conservation law restricts the entropy

• the entropy “flows” from the memory bit to the reservoir

Principle of Erasure:

01

0/1

E

T

0

1dE

0/1

E

T

work

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System:● spin ½ ½ particles● no B or E fields so spins states are energy degenerate● collisions between particles cause spin exchanges

0/1Memory bit: single spin ½ particle

Reservoir: collection of N spin ½ particles.

Possible states

,,

,,

Simple representation: , n

# of spin up

multiplicity (copy): 1,2,…

n particles are spin up

nN

21

21

Angular Momentum Cost

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0/1

zJ

Angular momentum diagram

states

Memory bit:

Reservoir:,n

0,11,1 , 1,2 , 1,3 ,

# of spin up

multiplicity (copy)

zJ

N

n1,2,…

21

21state

number of states with

12z J n N

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Reservoir as “canonical” ensemble (exchanging not energy)

Maximise entropy of reservoir

subject to

,

,, lnn

nn PP

1, 2reservoir

, 2

z nn

NJ P n N 1

,,

nnP&

Total is conserved

zJ

zJ

0,11,

zJ

0,1 1,

,n

Reservoir:Bigger spin bath:

,nP

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Reservoir as “canonical” ensemble (exchanging not energy)

Maximise entropy of reservoir

subject to

,

,, lnn

nn PP

NN

nPJn

nz 21

,,reservoir 2

1

,,

nnP&

Total is conserved

Jz

zJ

0,11,

zJ

1,0 1,

,n

Reservoir:Bigger spin bath:

121zJ

10 1 1

Average spin

1

1

2

2Z

Z

J

Je

,1

n

n N

eP

e

1 1ln

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0/1

Erasure protocolReservoir:

zJ 2

1P

2

1P

Memory spin:

zJ

0,11,

,1

n

n N

eP

e

1 1ln

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zJ

0,11,

0/1

Reservoir:

zJ

Coupling

0,1 1,1

1

eP

e

1

1P

e

Memory spin:Erasure protocol

,1

n

n N

eP

e

1 1ln

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Reservoir:

0/1

Increase Jz using ancilla in

memory(control)

ancilla (target)

zJ

2

this operation costs

Memory spin:

and CNOT operation

2,

zJ

0,12

Erasure protocol

,1

n

n N

eP

e

1 1ln

1

eP

e

1

1P

e

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2,

zJ

0,12

0/1

Reservoir:

zJ

2

2

21

eP

e

2

1

1P

e

2

Coupling

0,1 2,1


,1

n

n N

eP

e

1 1ln

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zJ

0,1

m

,1m

0/1

Reservoir:

,1

n

n N

eP

e

1 1ln

zJ

m

0 P

1 P

m

Repeat

Final state of memory spin & ancilla

memory erased ancilla in initial state


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zJ

1,0

m

,1m

Reservoir:

Nn

ne

eP

1,

1 1ln

m

Repeat

Final state of memory spin & ancilla

memory erased ancilla in initial state

0/1

zJ

Memory spin:

m

1P

2/

0 P

Total cost:The CNOT operation on state of memory spin consumes angular momentum. For step m:

1

m

m

eP

e

0 0 1

m

z mm m

eJ P

e

memory (m-1) mth ancilla

mth ancilla

m=0 term includes cost of initial state

ln 2z

J

1

1

2

2Z

Z

J

Je

Erasure protocol

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Single thermal reservoir: - used for both extraction and erasure

Impact

Q

erased memorywork

work

Q

heat engine

cycle

entropy

No net gain

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cycle

Two Thermal reservoirs:

- one for extraction, - one for erasure

Q1

work

entropy

increased entropy

Net gain if T1 > T2

T1

T2

Q2

work

erased memory &Q energy decreaseheat engine

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spin reservoir

zJ

,11,0

cycleentropy

Here:Thermal and Spin reservoirs:

- extract from thermal reservoir- erase with spin reservoir

spin

Q

workerased

memory &Q energy decrease

zJ

increased entropy

Gain if T1 > 0 heat engine

T1

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zJ

,11,0

Shannon

cost work

entropy

E

thermal reservoir

spin reservoir New

mechanism:

2nd Law Thermodynamics

Kelvin-Planck

It is impossible for a heat engine to produce net work in a cycle if it exchanges heat only with bodies at a single fixed temperature.

S 0 Schumacher (yesterday) “There can be no physical process whose sole effect is the erasure of information”

applies to thermal reservoirs only

Shannon entropy

general

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▀ cost of erasure depends on the conservation law

▀ thermal reservoir is a resource for erasure:

cost is

▀ spin reservoir is a resource for erasure:

cost is

ln 2ln 2E kT

where

kT

1

ln 2J

z

1 1ln

where

▀ 2nd Law is obeyed: total entropy is not decreased

▀ New mechanism

Summary

zJ

,11,0

Shannon

cost work

entropy

E

thermal reservoir

spin reservoir

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Spinning as a resource…

xkcd.com

erasure of information under conservation laws

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